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Full text of "Mathematical And Physical Papers - Iii"

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15.   The form of the equations of condition (30) points out sin2<9 as a factor of P, and since the  operation sin 6 -^ -r^ -^
performed on the function sin20 reproduces the same function with a coefficient - 2; it will be possible to satisfy equations (27) and (28) on the supposition that sin2# is a factor of ^ and ^2#. Assume then
Patting for convenience
n*f^l=p'm*.; ......................... (32),
and substituting in (27) and (28), we get
/1-^/1W = 0 ........... . ............ (33),
/2-^/2W-m2/2(r} = 0   .................. (34).
The equations of condition (30), (31) become, on putting /(r)
(35), (36).
We may obtain p  from  (29)  by putting for  ^  its  value (r), replacing after differentiation 2/1(r) by its equiva-
* When this operation is performed on the function sin0 dYJdO, the function is
reproduced with a coefficient - i (i + 1).    F4 here denotes a Laplace's function of the
itk order, which contains only one variable angle, namely 6.   Now \f/ may be ex-
panded in a series of quantities of the general form sm6dYi/d6.   For, since we are
only concerned with the differential coefficients of ^ with respect to r and 0, we have
a right to suppose i// to vanish at whatever point of space we please.   Let then
;//=0 when r=a and 0 = 0.    To find the value of ^ at a distance r from the origin,
along the axis of x positive, it will be sufficient to put 0 = 0, dd = Q in (20), and
integrate from r  a to r, whence i// = 0.    To find the value of i// at the same dis-
tance r along the axis of x negative, it will be sufficient to leave r constant, and
integrate d$ from. 0 = 0 to 0 = 7T.   Referring to (26), we see that the integral vanishes,
since the total flux across the surface of the sphere whose radius is r must be equal
to zero.   Hence ^ vanishes when 0 = 0 or =TT, and it appears from (26) that when
6 is very small or very nearly equal to x, ^ varies ultimately as sin2 0 for given
values of r. and t.   Hence ^ cosec 0, and therefore f\j/ cosec 6 dd, is finite even when
sin 0 vanishes, and therefore/^ cosec 0 dd may be expanded in a series of Laplace's
functions, and therefore i// itself in a series of quantities of the form sin0
It was somewhat in this way that I first obtained the form of the function ^.